The Nilpotency Class of Fitting Subgroups of Groups with Basis Property (original) (raw)

A group G is called group with basis property if for any subgroup H of G there exists a basis, i.e. a minimal generating set such that any two bases of H are equivalent. We show that any group with basis property is Frobenies group with kernel p-group P = F it(G) and complement q-group < y > of order q b. Let G be group with basis property, then F it(G) in general has no upper bound, but in some special cases the nilpotency class of F it(G) is bounded and that depends on the order of the complement group. We proved the following theorem: Let G be a finite group with basis property which is not a p−group, then the nilpotency class of the Fitting subgroup F it(G) has no upper bound.Moreover, for each prime number q > 2there exists a nite group with the basis property such that: c(F it(G) = q − 1; | < y > | = q.